Rational Numbers Class 8 Notes: CBSE Maths Chapter 1

1. Rational Numbers are numbers in the form of $\dfrac{p}{q}$ such that $q>0$. It is denoted by “Q”.

2. If the numerator and denominator are coprime and $q>0$ then the Rational Number is of the standard form.

3. Types of Rational Numbers:

i. Positive Rational Numbers: The sign of both the numerator and denominator are the same, i.e., either both are positive or both are negative. Ex: $\dfrac{2}{3},\dfrac{-7}{-8},...$

ii. Negative Rational Numbers: The sign of both the numerator and denominator are the same, i.e., if the numerator is negative, the denominator will be positive. Similarly, if the numerator is positive, the denominator is negative. Ex: $\dfrac{2}{-3},\dfrac{-7}{8},...$

iii. Zero Rational Numbers: The numerator is always zero. Ex: $\dfrac{0}{3},\dfrac{0}{8},...$

4. Properties of Rational Numbers:

4.1 Closure Property

i. Whole number:

ii. Integers

iii. Rational numbers

To prove the closure property for rational numbers under addition, subtraction, multiplication, and division, let's consider two rational numbers. A rational number can be expressed in the form $\dfrac{a}{b} $, where $ a$ and $ b $ are integers, and $ b\neq 0 $.

 1. Addition

To prove that rational numbers are closed under addition:

Let $\dfrac{a}{b} $ and $\dfrac{c}{d} $ be two rational numbers.

$\dfrac{a}{b} + \dfrac{c}{d} $= $\dfrac{ad+bc}{bd} $ and$\dfrac{c}{d}+ \dfrac{a}{b} $= $\dfrac{cb+ad}{db} $

Adding them: 

  $\dfrac{a}{b} +\dfrac{c}{d} = \dfrac{ad + bc}{bd}$

Since the numerator $ ad + bc $ and the denominator $ bd $ are both integers (as integers are closed under addition and multiplication), the result is a rational number.

Conclusion: Rational numbers are closed under addition.

 2. Subtraction

To prove that rational numbers are closed under subtraction:

Let $\dfrac{a}{b} $ and $\dfrac{c}{d} $ be two rational numbers.

Subtracting them:

  $\dfrac{a}{b} -\dfrac{c}{d} =\dfrac{ad - bc}{bd}$

Since the numerator $ ad - bc $ and the denominator $ bd $ are both integers, the result is a rational number.

Conclusion: Rational numbers are closed under subtraction.

 3. Multiplication

To prove that rational numbers are closed under multiplication:

Let $\dfrac{a}{b} $ and $\dfrac{c}{d} $ be two rational numbers.

Multiplying them:

  $\dfrac{a}{b}\times\dfrac{c}{d} =\dfrac{ac}{bd}$

Since the numerator $ ac $ and the denominator $ bd $ are both integers, the result is a rational number.

Conclusion: Rational numbers are closed under multiplication.

 4. Division

To prove that rational numbers are closed under division (except when dividing by zero):

Let $\dfrac{a}{b} $ and $\dfrac{c}{d} $ be two rational numbers where $ c\neq 0 $ and $ d\neq 0 $.

Dividing them:

  $ \dfrac{a}{b}\div\dfrac{c}{d} =\dfrac{a}{b}\times\dfrac{d}{c} =\dfrac{ad}{bc}$

Since the numerator $ ad $ and the denominator $ bc $ are both integers and $ bc\neq 0 $, the result is a rational number.

Conclusion: Rational numbers are closed under division, except when dividing by zero.

Therefore, rational numbers are closed under addition, subtraction, multiplication, and division (with the exception of division by zero).

4.2 Commutative Property: 

i. Whole numbers

Ii. Integers

Iii. Rational numbers

Let's now check the commutative property for rational numbers under addition, subtraction, multiplication, and division.

 1. Addition (Commutative Property):

The commutative property of addition states that for any two numbers $ a $ and $ b $, the order of addition does not affect the result, i.e., $ a + b = b + a $.

Let $ \dfrac{a}{b} $ and $ \dfrac{c}{d} $ be two rational numbers.

We know:

$\dfrac{a}{b} + \dfrac{c}{d} $= $\dfrac{ad + bc}{bd} $ and $\dfrac{c}{d} + \dfrac{a}{b} $ = $\dfrac{cb + ad}{db} $

Since $ad + bc $= $bc + ad $ and $bd = db $, we conclude that:

$\dfrac{a}{b} + \dfrac{c}{d} $= $\dfrac{c}{d} + \dfrac{a}{b} $

Conclusion: Rational numbers are commutative under addition.

 2. Subtraction (Commutative Property):

The commutative property of subtraction states that for any two numbers $ a $ and $ b $, $ a - b $ does not necessarily equal $ b - a $.

- Let $ \frac{a}{b} $ and $ \frac{c}{d} $ be two rational numbers.

We know:

  $\dfrac{a}{b} - \dfrac{c}{d} = \dfrac{ad - bc}{bd}$ and $\dfrac{c}{d} $ - $\dfrac{a}{b} $= $\dfrac{cb - ad}{db} $

Since $ ad - bc \neq cb - ad $, we conclude:

$\dfrac{a}{b} $ -$\dfrac{c}{d} \neq \dfrac{c}{d} $ - $\dfrac{a}{b} $

Conclusion: Rational numbers are not commutative under subtraction.

 3. Multiplication (Commutative Property):

The commutative property of multiplication states that for any two numbers $ a $ and $ b $, $ a \times b = b \times a $.

Let $\dfrac{a}{b} $ and $\dfrac{c}{d} $ be two rational numbers.

We know:

$\dfrac{a}{b} \times \dfrac{c}{d} $ = $\dfrac{ac}{bd}$ and $\dfrac{c}{d} \times $ $\dfrac{a}{b} $ = $\dfrac{ca}{db}$

Since $ ac = ca $ and $ bd = db $, we conclude:

$\dfrac{a}{b} \times \dfrac{c}{d} $= $\dfrac{c}{d} \times $ $\dfrac{a}{b} $

Conclusion: Rational numbers are commutative under multiplication.

 4. Division (Commutative Property):

The commutative property of division states that for any two numbers $ a $ and $ b $, $ a \div b $ does not necessarily equal $ b \div a $.

Let $\dfrac{a}{b} $ and $\dfrac{c}{d} $ be two rational numbers where $ c \neq 0 $ and $ d \neq 0 $.

We know:

$\dfrac{a}{b} \div \dfrac{c}{d} $ = $\dfrac{ad}{bc} $ and $\dfrac{c}{d} \div \dfrac{a}{b} $ = $\dfrac{cb}{ad} $

Since $ ad \neq cb $ and $ bc \neq ad $, we conclude:

$\dfrac{a}{b} \div \dfrac{c}{d}$ $ \neq $ $\dfrac{c}{d} \div \dfrac{a}{b}$

Conclusion: Rational numbers are not commutative under division.

 Summary of Commutative Property for Rational Numbers:

- Addition: Commutative.

- Subtraction: Not commutative.

- Multiplication: Commutative.

- Division: Not commutative.

4.3. Associative Property: 

i. Whole numbers

ii. Integers

iii. Rational Numbers

Let's briefly check the associative property for rational numbers under addition, subtraction, multiplication, and division.

 1. Addition (Associative Property):

The associative property of addition states that the grouping of numbers does not affect the sum, i.e., $ (a + b) + c = a + (b + c) $.

Let $ \dfrac{a}{b} $, $\dfrac{c}{d} $, and $\dfrac{e}{f} $ be three rational numbers.

We know:

  $\left( \dfrac{a}{b} + \dfrac{c}{d} \right) + \dfrac{e}{f} $ 

= $\dfrac{ad + bc}{bd} + \dfrac{e}{f} $ = $\dfrac{(ad + bc)f + bde}{bdf} $ and 

$\dfrac{a}{b} + \left( \dfrac{c}{d} + \dfrac{e}{f} \right) $= $\dfrac{a}{b} $ + $\dfrac{cf+de}{df} $ = $\dfrac{(cf + de)b + adf}{bdf} $

Both are equal, so:

$ (a + b) + c = a + (b + c) $

Conclusion: Rational numbers are associative under addition.

 1. Distributive Property for Integers

Let $ a = 3 $, $ b = -2 $, and $ c = 4 $.

\[3 \times (-2 + 4) = 3 \times 2 = 6\]

On the other hand:

\[(3 \times -2) + (3 \times 4) = -6 + 12 = 6\]

Thus, $ 3 \times (-2 + 4) = (3 \times -2) + (3 \times 4) $.

- Conclusion: The distributive property holds for integers.

 2. Distributive Property for Whole Numbers

Let $ a = 2 $, $ b = 5 $, and $ c = 3 $ (all whole numbers).

\[2 \times (5 + 3) = 2 \times 8 = 16\]

On the other hand:

\[(2 \times 5) + (2 \times 3) = 10 + 6 = 16\]

Thus, $ 2 \times (5 + 3) = (2 \times 5) + (2 \times 3) $.

- Conclusion: The distributive property holds for whole numbers.

 3. Distributive Property for Rational Numbers

Let $ a = \frac{1}{2} $, $ b = \frac{3}{4} $, and $ c = \frac{5}{6} $.

\[\frac{1}{2} \times \left( \frac{3}{4} + \frac{5}{6} \right) = \frac{1}{2} \times \left( \frac{9}{12} + \frac{10}{12} \right) = \frac{1}{2} \times \frac{19}{12} = \frac{19}{24}\]

On the other hand:

\[\left( \frac{1}{2} \times \frac{3}{4} \right) + \left( \frac{1}{2} \times \frac{5}{6} \right) = \frac{3}{8} + \frac{5}{12} = \frac{9}{24} + \frac{10}{24} = \frac{19}{24}\]

Thus, $ \frac{1}{2} \times \left( \frac{3}{4} + \frac{5}{6} \right) = \left( \frac{1}{2} \times \frac{3}{4} \right) + \left( \frac{1}{2} \times \frac{5}{6} \right) $.

- Conclusion: The distributive property holds for rational numbers.

Final Summary:

- The distributive property applies to integers, whole numbers, and rational numbers.

- It shows that multiplying a number by a sum (or difference) is the same as multiplying the number by each addend (or subtrahend) and then adding (or subtracting) the results.

v. General Properties: 

• A rational number can be a fraction or not, but vice versa is true.

• Rational numbers can be denoted on a number line.

• There is $'n'$ number of rational numbers between any two rational numbers.

5. Role of Zero: Also known as the Additive Identity

Whenever $'0'$ is added to any rational number, the answer is the Rational number itself.

Ex: If $'a'$ is any rational number, then $a+0=0+a=a$

6. Role of One: Also known as the Multiplicative Identity.

Whenever $'1'$ is multiplied by any rational number, the answer is the Rational number itself.

Ex: If $'a'$ is any rational number, then $a\times 1=1\times a=a$

7. Additive Inverse:

The Additive Inverse of any rational number is the same rational number with the opposite sign. The additive inverse of $\dfrac{a}{b}$ is $-\dfrac{a}{b}$. Similarly, the additive inverse of $-\dfrac{a}{b}$ is $\dfrac{a}{b}$, where $\dfrac{a}{b}$ is the rational number.

8. Multiplicative Inverse: Also known as the Reciprocal.

The Multiplicative Inverse of any rational number is the inverse of the same rational number. The multiplicative inverse of $\dfrac{a}{b}$ is $\dfrac{b}{a}$. Similarly, the multiplicative inverse of $\dfrac{b}{a}$ is $\dfrac{a}{b}$, where $\dfrac{a}{b}$ and $\dfrac{b}{a}$ is any rational number.

Important Formulas of Class 8 Chapter 1 Maths Rational Numbers You Shouldn’t Miss!

These formulas are key for performing operations with rational numbers and will help you solve various problems in this chapter.

1. Definition of Rational Numbers: A rational number is any number that can be expressed as a fraction $\dfrac{p}{q}$, where $p$ and $q$ are integers, and $q \neq 0$.

2. Addition of Rational Numbers: To add two rational numbers $\dfrac{a}{b}$ and $\dfrac{c}{d}$, use:

   $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{a \cdot d + b \cdot c}{b \cdot d} $

3. Subtraction of Rational Numbers: To subtract $\dfrac{a}{b}$ from $\dfrac{c}{d}$, use:

   $\dfrac{c}{d} - \dfrac{a}{b} = \dfrac{c \cdot b - a \cdot d}{b \cdot d}$

4. Multiplication of Rational Numbers: To multiply two rational numbers $\dfrac{a}{b}$ and $\dfrac{c}{d}$, use:

   $\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \cdot c}{b \cdot d}$

5. Division of Rational Numbers: To divide $\dfrac{a}{b}$ by $\dfrac{c}{d}$, use:

   $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} $ = $\dfrac{a \cdot d}{b \cdot c} $

6. Reciprocal of a Rational Number: The reciprocal of $\dfrac{a}{b}$ is $\dfrac{b}{a}$, provided $a \neq 0$.

Importance of Chapter 1 Rational Numbers Class 8 Notes

• Foundation for Algebra: Understanding rational numbers is crucial for grasping more advanced algebraic concepts. This chapter lays the groundwork for working with equations and inequalities.

• Basic Arithmetic Operations: Mastery of addition, subtraction, multiplication, and division of rational numbers is essential for solving various mathematical problems and performing operations accurately.

• Fractions and Decimals: Rational numbers include fractions and decimals, which are frequently used in real-life scenarios. This chapter helps in converting between fractions, decimals, and percentages.

• Problem-Solving Skills: The chapter enhances problem-solving skills by teaching how to handle rational numbers in different contexts, which is useful for tackling complex problems in higher grades.

• Preparation for Exams: Comprehensive notes on rational numbers provide a clear understanding and practice opportunities, aiding in effective exam preparation and boosting overall performance.

Tips for Learning the Class 8 Maths Chapter 1 Rational Numbers

1. Understand the Basics: Start by understanding what rational numbers are—numbers that can be expressed as fractions $\dfrac{p}{q}$​ where p and q are integers, and q≠0.

2. Practice Operations: Work on addition, subtraction, multiplication, and division of rational numbers. Practice with different examples to become comfortable with the operations.

3. Convert Between Forms: Learn how to convert between fractions, decimals, and percentages. This skill is essential for solving a variety of problems.

4. Use Visual Aids: Draw number lines or use fraction bars to visualize how rational numbers compare and how operations affect them. Visual aids can help make abstract concepts more concrete.

5. Solve Practice Problems: Regularly solve practice problems from your textbook or online resources. This will reinforce your understanding and help you apply the concepts effectively.

6. Work on Word Problems: Practice solving word problems involving rational numbers to improve your problem-solving skills and learn how to apply concepts in real-life situations.

Conclusion

Chapter 1 of Class 8 Maths on Rational Numbers provides essential knowledge for understanding and working with fractions, decimals, and percentages. Mastery of this chapter is fundamental for performing arithmetic operations, solving problems, and preparing for more advanced mathematical concepts. By focusing on practice, using visual aids, and reviewing key formulas, you can strengthen your grasp of rational numbers. These skills will not only aid in academic success, but also in applying mathematical concepts to real-life situations. Use the notes and tips provided to enhance your learning and ensure a solid foundation in rational numbers.

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